Wild Harmonic Bundles and Wild Pure Twistor D-Modules

Contents
1. Introduction
1.1. Contents of this monograph
1.2. Asymptotic behaviour of wild harmonic bundles
1.3. Application to meromorphic flat bundles
1.4. Application to holonomic ?-modules and wild pure twistor ?-modules
1.5. Acknowledgement
Part I. Good Meromorphic ?-Flat Bundles
2. Good Formal Property of Meromorphic ?-Flat Bundle
2.1. Good set of irregular values and truncations
2.2. Good lattice in the formal case
2.3. Good lattice of meromorphic ?-flat bundle
2.4. Decompositions
2.5. Good filtered ?-flat bundle
2.6. Good lattice at the level ?
2.7. Good Deligne-Malgrange lattice and Deligne-Malgrange lattice
2.8. Family of filtered λ-flat bundles and KMS-structure
3. Stokes Structure of Good ?-Meromorphic Flat Bundle
3.1. Preliminaries
3.2. Good meromorphic ?-flat bundle
3.3. Good meromorphic ?-flat bundle at the level ?
3.4. Some notation
3.5. Preliminary in the smooth divisor case
3.6. Proof of the statements in Section 3.3
3.7. Proof of the statements in Section 3.2
4. Full Stokes Data and Riemann-Hilbert-Birkhoff Correspondence
4.1. Full pre-Stokes data
4.2. Full Stokes data
4.3. Riemann-Hilbert-Birkhoff correspondence
4.4. Extension of Stokes data
4.5. Deformation
5. ?^2-Cohomology of Filtered λ-Flat Bundle on Curves
5.1. Local quasi-isomorphisms for fixed λ
5.2. Proof for fixed λ
5.3. Local quasi-isomorphisms in family
5.4. Proof in the family case
6. Meromorphic Variation of Twistor Structure
6.1. Variation of polarized pure twistor structure
6.2. Good meromorphic prolongment of variation of twistor structure
Part II. Prolongation of Wild Harmonic Bundle
7. Prolongments ?ℰ^λ for Unramifiedly Good Wild Harmonic Bundles
7.1. Definition of wild harmonic bundles
7.2. Simpson’s main estimate and acceptability of the associated bundles
7.3. Proof of the estimates
7.4. The associated good filtered λ-flat bundle
7.5. Comparison of the irregular decompositions in the case ? < 0_ℓ
7.6. Small deformation of ?ℰ^λ in the smooth divisor case
7.7. Boundedness of some section
8. Some Basic Results in the Curve Case
8.1. Norm estimate for holomorphic sections of ?_? ℰ^λ
8.2. Comparison of the data at ?
8.3. A characterization of the lattices for generic λ
8.4. Harmonic forms
9. Associated Family of Meromorphic λ-Flat Bundles
9.1. Filtered bundle ((?_∗)^(λ_0)) ℰ
9.2. Family of meromorphic flat λ-connections on ((?_∗)^(λ_0)) ℰ
9.3. Estimate of the norms of partially flat sections
9.4. Locally uniform comparison of the irregular decompositions
10. Smooth divisor case
10.1. Acceptability of (?^(λ_0)) ℎ
10.2. Locally uniform comparison of the irregular decompositions
10.3. Norm estimate for a fixed λ
10.4. Norm estimate in family
11. Prolongation and reduction of variations of polarized pure twistor structures
11.1. Canonical prolongation
11.2. Reduction and uniqueness
11.3. Preliminary for convergence
11.4. One step reduction
11.5. Full reduction in the smooth divisor case
11.6. End of Proof of Theorem 11.2.2
11.7. Norm estimate
11.8. Regular meromorphic variation of twistor structure on a disc (Appendix)
12. Prolongation as ℛ-triple
12.1. ℛ-module associated to unramifiedly good wild harmonic bundle
12.2. Review of some results in the tame case
12.3. The case where Irr(?) consists of only one element
12.4. Specialization of the associated ℛ_?-module
12.5. Construction of the ℛ-triple (?,?,ℭ)
12.6. A characterization of the prolongment in the one dimensional case
12.7. Specialization of the associated ℛ-triples
12.8. Prolongation of ramified wild harmonic bundle on curve
Part III. Kobayashi-Hitchin correspondence
13. Preliminaries
13.1. Preliminaries for μ_?-polystable filtered flat sheaves
13.2. Mehta-Ramanathan type theorem
13.3. Auxiliary metrics
13.4. Harmonic bundles on curves
13.5. Some characterizations of wildness of harmonic bundle
13.6. The filtered flat bundle associated to wild harmonic bundle
13.7. Perturbation
14. Construction of an initial metric and preliminary correspondence
14.1. Around a crossing point
14.2. Around smooth point
14.3. Some formulas for the parabolic Chern character
14.4. Preliminary correspondence
14.5. Bogomolov-Gieseker inequality
15. Preliminaries for the resolution of turning points
15.1. Resolution for a tuple of ideals
15.2. Separation of the ramification and the polar part of a Higgs field
15.3. Resolution for generically good Higgs field
16. Kobayashi-Hitchin correspondence and some applications
16.1. Kobayashi-Hitchin correspondence for good filtered flat bundles
16.2. Applications to algebraic meromorphic flat bundles
16.3. Proof of Theorems 16.2.1 and 16.2.4
16.4. Minor refinement of the result in Section 16.2
Part IV. Application to Wild Pure Twistor ?-Modules
17. Wild pure twistor ?-modules
17.1. Review of wild pure twistor ?-modules due to Sabbah
17.2. Wild harmonic bundles and wild pure twistor ?-modules on curves
17.3. Gysin map for wild pure twistor ?-modules
18. The Hard Lefschetz Theorem
18.1. Statement
18.2. Step 1
18.3. Step 2
18.4. Step 3
19. Correspondences
19.1. Wild harmonic bundles and wild pure twistor ?-modules
19.2. Prolongation to polarized wild pure twistor ?-modules
19.3. Wildness and uniqueness
19.4. Application to algebraic semisimple holonomic ?-modules
Part V. Appendix
20. Preliminaries from analysis on multi-sectors
20.1. Hukuhara-Malmquist type theorem
20.2. Estimates of some integrals on a sector
20.3. Some Estimates on a multi-sector
21. Acceptable bundles
21.1. Some general results on vector bundles on Kähler manifolds
21.2. Twist of the metric of an acceptable bundle
21.3. Prolongation to filtered bundle
21.4. One dimensional case
21.5. Extension of holomorphic sections I
21.6. Extension of holomorphic sections II
21.7. Proof of Theorems 21.3.1 and 21.3.2
21.8. Small deformation
21.9. Complement
22. Review on ℛ-modules, ℛ-triples and variants
22.1. Filtered rings
22.2. ℛ-modules
22.3. Specialization of ℛ-modules
22.4. ℛ_? (∗?)-modules
22.5. Formal ℛ-modules
22.6. Preliminaries for ?-modules
22.7. Complement
22.8. The sheaves ??_(?×?/?) and (??_(?×?/?))^mod
22.9. ℛ-triples
22.10. Specialization of ℛ-triples
22.11. ℛ (∗?)-triples
22.12. Comparison
Bibliography
Index